This parametric grapher draws Cartesian parametric curves represented by p(t) = [f(t),g(t)], and polar parametric curves represented by p(t) = [r(t),θ(t)] using a sophisticated and easy-to-follow animation algorithm. This capability helps visualize the construction of parametric curves from beginning to end.
You can pause, resume, and control the speed of parametric curve plotting process with the provided slider.
Besides being the only polar parametric curve grapher available online, it also has the unique feature of rotating radial axes when constructing polar parametric curves from scratch.
You can instantly visualize parametric curves in either coordinate system by toggling the Polar checkbox.
Additionally, this parametric grapher allows you to rotate axes and graph in oblique coordinate systems, where axes can be rotated to any angle and have any orientation.
Note: Our graphing software allows you to use constant expressions, such as π or 1+√(2) wherever you can use a literal number.
Instructions for Parametric Curve Grapher (Cartesian and Polar)
This interactive parametric curve grapher has been developed to specifically show how a parametric curverepresented by p(t)=[f(t),g(t)] is graphed in both Cartesian and polar coordinate systems using animation, ideal for teaching or learning about the process of parametric graphing.
Remark: It is customary to represent a parametric expression as either [x(t),y(t)] or [r(t),θ(t)] depending on whether graphing them in Cartesian or polar coordinate systems, respectively.
To explore parametric graphs, type a parametric expression in any expression box, for example, [sin(t),cos(t)]—the use of the enclosing brackets [ ] is optional. The parametric equations plotter graphs as you type (default) in the selected Cartesian or polar coordinate systems.
Our parametric grapher appends a suitable interval (domain) to parametric expressions and graphs them on the specified domain. You can change the end points, but they must be finite for parametric graphing. The parametric grapher automatically changes infinite values to finite values.
This parametric graphing calculator uses a unique animation algorithm to visualize the step-by-step construction of Cartesian and polar parametric graphs like no other grapher.
This parametric grapher allows you to watch the entire animated parametric graphing process, as detailed below.
Using Parametric Curve Graphing Animation Feature
This feature allows you to visualize the step-by-step process of creating a parametric graph. To activate the animation feature, press ► at the bottom of the grapher (if hidden, press Animate first).
The parametric grapher starts the parametric graphing animation for the focused parametric expression The animation progressively draws the parametric curve, starting at t1 and ending at t2. It shows whether any loops or parts of the parametric curve are traced multiple times.
Controlling Animation
To pause the animation, press ‖.
To stop the animation, press Done. This also closes the animation interface. To display the animation interface again, press the Animate button.
Use the slider to adjust the animation speed.
Additional Options
You can optionally show or hide the rotating radial axes by checking or unchecking the Show radial axes checkbox (by default, it's checked).
Graphing Multiple Parametric Curves
To graph multiple parametric curves, press the » button to show the multi-graph pane with expression panels.
Add or delete panels with the + or × buttons if necessary.
Select or deselect checkboxes to show or hide graphs.
Remark: To graph piecewise defined parametric curves type in each piece with the corresponding subinterval as a single parametric expression.
Graph Accuracy Setting
The quality of the resulting parametric graph is controlled by the Graph Fineness setting. A higher graph fineness results in a more accurate graph, but it also takes longer to graph the parametric curve.
Labeling Axes
To label an axis, click on the ▼ icon at the top right of the canvas. Type in any number for which you want to label the axes, and press the Label button. You can also use constants like π, or even constant expressions such as 1+3π/2.
Rotating Axes
Our graphing software offers a unique feature: axis rotation. This allows you to visualize the graph of a given parametric curve in the oblique (non-perpendicular) coordinate system. To rotate an axis, click on the ▼ icon at the top right of the canvas, and then enter the angles by which you want to rotate an axis and press the Rotate button. The axis will rotate and the graphs will be redrawn to reflect the rotation.
Copying & Saving Graphs
Click the Copy/Save graph button. This will create a copy of all the graphs on the canvas (graphing area) which will appear as an image below the parametric graphing calculator.
Right-click on the image and select the appropriate option from the context menu. Depending on your device and preferences, you might be able to:
Copy: Create a copy of the image to your clipboard for pasting elsewhere.
Save image as... : Save the image as a file on your device in a chosen format (e.g., PNG, JPG).
Evaluating Parametric Equations
To evaluate a parametric expression, type a number or constant expression in the provided box.
The parametric graphing calculator will display the calculated value, rounded to the number of decimal places set by the slider.
Interesting Curves
Interesting curves: Graph any of the expressions under Interesting Graphs. For best results, you may need to select Graph Fineness as "+1" or higher.
Mouse Operations in Our Graphing Software
Our graphing software allows you to use your mouse to perform unique operations as outlined below.
How to Rotate Axes
To rotate an axis, hold down the Alt key and click on or near the axis. This will select the axis, and its color will change to red. Move the mouse. The selected axis will rotate accordingly, and the graphs are redrawn to reflect the rotation of the axis. Click again with the Alt key pressed, to release the axis.
To restore the coordinate system to its normal state press the Reset button (if hidden, press the ▼ icon).
How to Change Scales
To change the scale (zoom in one direction), hold down the ctrl key and click on an axis. The point that was clicked will be labeled "1" if clicked on the positive side, or "-1" if clicked on the negative side of the axis; it becomes the new unit for that axis.
How to Translate the Coordinate system
You can move the coordinate system in one of the following ways:
Click on the canvas and drag the mouse; this will translate the origin and the graphs together.
Double-click on the canvas to move the origin to the location where you clicked; the graphs are updated.
Settings
Press the ⚙ (gear) button to set options (if the button is hidden, first click on the ▼ icon at the top right of the canvas):
Change graphthickness using the slider.
Select the angle mode (radians - default, degrees, grades).
If you deselect the Graph as you type option, you will have to press Graph selected expressions, which then appears at the bottom of the calculator, to update the graphs whenever you make any changes to the expressions or the coordinate plane (i.e., move the origin, rotate axes, etc.).
You have the option to display controls that will automatically rotate the axes.
The parametric equations graphing calculator remembers the expressions so that you can use them in future visits. You can press Reset Calculator to clear them.
How the Parametric Grapher Works
Understanding how a parametric curve is formed in both Cartesian and polar coordinate systems is greatly enhanced by observing its step-by-step construction. This is especially true for complex parametric curves, which often exhibit loops and intricate patterns. Almost all other parametric graphers, including leading online graphing calculators, present the final curve without revealing its formation, making it impossible to follow its path or identify loops.
Our parametric grapher offers a comprehensive approach by employing Cartesian and polar coordinate systems and incorporating a dynamic animation feature. This animation visually demonstrates the curve's construction, allowing users to control the animation speed, pause, and resume it for clear observation. By watching the curve unfold step-by-step, users can gain insights into the behavior of even the most intricate parametric curves.
Designed to provide an in-depth understanding of parametric curves, our interactive graphing tool showcases the curve's creation process in both coordinate systems. Starting from an initial value t1, the graph is progressively built to the final value t2, clearly indicating any loops or retraced portions.
In particular, this unique polar parametric curve grapher works similarly to our polar function grapher (how?), except that it locates each point (r,θ) by calculating both r and θ as functions of t.
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